—If a function is analytic on an interval, then the function is expressed as the Taylor expansion about a point in the interval. Furthermore, possibility of Taylor expansions of functions about two or three points has also been studying as useful expressions in several fields of mathematical sciences. In this paper, we show the following main result by estimating values of divided differences: Let f is a polynomial p on the interval [├ 1⁄3,∞) ┤ and f is a polynomial q on the interval (├ -∞,1⁄3] ┤. Then, we show that f is expressed as the two point Taylor expansion about -1,1 with the multiplicity weight (2,1) on the interval (α,β), where α is the solution of (x+1)^2 (x-1)=-32⁄27 with α<-1 and β is the solution of (x+1)^2 (x-1)=32⁄27 with β>1.
—Polynomial interpolation, Hermite interpolation, Taylor expansion, two point taylor expansion.
The authors are with the School of Science and Technology, Kwansei Gakuin University, Japan (e-mail: email@example.com).
Cite: Kazuaki Kitahara and Taka-Aki Okuno, "A Note on Two Point Taylor Expansion III," International Journal of Modeling and Optimization vol. 4, no. 4, pp. 95-99, 2014.